Geometry · Virginia SOL released test, spring 2015 · Day 1

The grammar: if, then

Every theorem you will meet this month is written in one sentence shape. If two lines are parallel, then alternate interior angles are congruent. If a triangle has two congruent sides, then the angles opposite them are congruent. Learn to take that sentence apart and put it back together, and you can read the whole subject. That is today.

Text in this sans face is the test's own wording, word for word. Text in this serif is ours. Anything with a copper rule on its left is a problem. The three doors under a problem stay shut until somebody opens them.

BEFORE 0:00 · TUTOR

Before we start

Press Start in the strip at the bottom when the hour begins; the clock turns red at 60. Press T (or the Tutor button) to show your layer — the red notes carry the answer, what each wrong choice means, the check for that item, and where to go if it is missed. Turn it off before you share the screen. Notes opens a pad that saves itself in this browser; Stamp drops the clock time into it.

Tutor layer is onYou are seeing what the student does not. Everything in a red panel is yours. The button under each item, mark satisfied, is for when the student says it right out loud and typing it would only slow you down.
0:00 – 0:08

Placement — three questions, no teaching

One question from each of the test's three parts. Nothing is explained yet and nothing is marked; the point is to find out where you already are, so the month starts from the right place. Answer each one and we move on.

SayThese are not a quiz. I want to see how you think before I say anything, so that I do not waste your time on what you already have.
SOL 2015 · #2 Reasoning, Lines, and Transformations

The graph of line j is shown. Which ratio represents the slope of a line parallel to line j?

-4-3-2-1 1234 4321 -1-2-3-4 xy (−4, −4)(3, −1)j
Noted. Next one.
#2 · key ASlope = rise/run = (−1 − (−4)) / (3 − (−4)) = 3/7; parallel keeps the slope.
  • B 5/7 — miscounted the rise on the grid.
  • C 7/5 — inverted, and miscounted.
  • D 7/3 — run over rise: inverted the fraction.
If missed → the Slopes ∥ ⊥ move (Day 3). Do not fix it now.
SOL 2015 · #24 Triangles · type-in

Select the measures that could be the three side lengths of a right triangle.

left legbottom leghypotenuse
left bottom hypotenuse
Noted. Next one.
#24 · key 20, 21 on the legs · 29 on the hypotenuse20² + 21² = 400 + 441 = 841 = 29². The largest number must sit on the hypotenuse; 17 belongs nowhere.
  • 17 used anywhere — did not test a² + b² = c²; picked by size or by feel.
  • 29 on a leg — has the theorem, does not have the hypotenuse rule.
  • Right set, wrong slot for the legs — fine; the two legs are interchangeable.
If missed → the Pythagorean move (Week 2). Do not fix it now.
SOL 2015 · #32 Polygons, Circles, and Three-Dimensional Figures · type-in

The height and radius of a cone are each multiplied by 3. What effect does this have on the volume of the cone? The volume of the cone is multiplied by —

Noted. That is the placement done.
#32 · key 27Volume carries three lengths, so a factor of 3 on every length is 3 × 3 × 3.
  • 3 — scaled once: treated volume like a length.
  • 9 — scaled twice: area thinking, or squared the radius and forgot the height was scaled too.
  • 6 — added the two 3s.
If missed → the Scaling move (Week 3). This is also the cleanest example on the whole test of a check that must run before the answer: "what dimension is this quantity?"
0:08 – 0:12

The map

The test has forty-five questions in three parts, and the parts are in an order that means something. Reasoning, Lines, and Transformations comes first because the other two are built out of it. Triangles come second because every polygon, every circle problem with a chord in it, and every scale-model question is a triangle wearing a coat. Polygons, Circles, and Three-Dimensional Figures come last because they need both of the others.

So we start at the root. Not because it is easy — some of it is the hardest thinking on the test — but because when something in week three goes wrong, the fix is almost always back here.

SayIf you get stuck later in the month, I am not going to give you more of the same problem. I am going to walk you back to the thing that feeds it. Today is the thing that feeds everything.
Sheng, said plainlyMother feeds child. Treat the mother. The plan page carries the full map of which move feeds which; today's move has no mother — if it is shaky, slow down here rather than pushing on.
0:12 – 0:20

If, then

A conditional statement has two parts. The part after if is the hypothesis. The part after then is the conclusion. The statement claims that whenever the hypothesis is true, the conclusion is true too. It claims nothing else.

Take the sentence the test uses in question 1: If a triangle is equilateral, then it is isosceles. The hypothesis is

That is the conclusion — the part after then. The hypothesis is the part after if.

Good. Now the move that the test loves more than any other. Take the two parts and swap them. You get a new statement: If a triangle is isosceles, then it is equilateral. That is the converse. Is it true?

Draw one. A triangle with sides 5, 5, 8 is isosceles and not equilateral. The converse is a new claim, and this one happens to be false.

That is the whole lesson in one line, and it is worth saying slowly: swapping the parts makes a new statement, and the new statement has to earn its own truth. The original being true tells you nothing about the converse.

There are two more relatives. Negate both parts without swapping and you get the inverse: If a triangle is not equilateral, then it is not isosceles. Swap and negate and you get the contrapositive: If a triangle is not isosceles, then it is not equilateral. One of these three always agrees with the original — it is true exactly when the original is. Which?

The converse is the swap alone, and we just found a counterexample to it.The inverse is the converse in disguise — it is the converse's contrapositive — so it fails whenever the converse fails.

The contrapositive. If a triangle is not isosceles, it certainly is not equilateral. Keep that pairing: original ≡ contrapositive, and converse ≡ inverse. Two families, not four strangers.

SayEverything the test does with logic is one of four moves: read the parts, swap them, negate them, or chain two statements together. We are going to do all four in the next forty minutes.
Ke for this sectionThe control is: a reversed statement is a new statement. Watch for the student treating the converse as automatic. If they answer "Yes" to the converse being true, do not correct — ask for a triangle with sides 5, 5, 8 and let the picture do it.
0:20 – 0:32

Worked: the converse

Here is the test asking exactly that. We will do this one in five moves, and the five moves are the same five for every problem this month — so watch the shape, not just the answer.

SOL 2015 · #8 Reasoning, Lines, and Transformations

Which is the converse of the following statement?
If a⁄b = c, then a = bc.

  1. FireduplicateWrite the statement out yourself. Box the part after if. Box the part after then. Do not skip this; the boxes are the whole method.
  2. MetaltranscribeGive the boxes names. p: a⁄b = c. q: a = bc. The original is p → q.
  3. WoodrearrangeConverse means swap: q → p. In words: If a = bc, then a⁄b = c.
  4. EarthcollapseFour options collapse to one question: which one reads q → p with nothing negated? Only B.
  5. WatermeaningThe converse here happens to be true — multiply both sides of a = bc by 1⁄b. But it is true on its own account, not because the original was. That is luck, not logic.
Nudge

Which two parts does the converse swap — and does it touch the word not?

Check my answer

B

The work, one step at a time
  1. Name the parts. Hypothesis: a⁄b = c. Conclusion: a = bc.
  2. Swap them. New hypothesis: a = bc. New conclusion: a⁄b = c.
  3. Do not negate anything. The converse never introduces a not.
  4. Match. "If a = bc, then a⁄b = c" is option B.
#8 · key B
  • A inverse (negate both).
  • C contrapositive (swap and negate) — the "smart" wrong answer; a student who picks C usually knows more than one who picks A.
  • D negated the conclusion only.
Control: converse = swap, and nothing else. Mother: none — this is the root. If shaky, stay here and do the inline checks in §3 again with a different sentence.
0:32 – 0:42

Same control, in pictures

The test asks the converse question a second time on the very first page, and it disguises it as a drawing. This time the five moves are yours: the middle of the working is left out. Fill it.

SOL 2015 · #1 Reasoning, Lines, and Transformations

Which Venn diagram accurately represents the information in the following statement?
If a triangle is equilateral, then it is isosceles.

ATriangles IsoscelesEquilateral CTriangles EquilateralIsosceles BTriangles IsoscelesEquilateral DTriangles IsoscelesEquilateral
  1. FireduplicateBox the hypothesis and the conclusion in the sentence, same as before.
  2. MetaltranscribeTurn "if p then q" into set language: every p is a q. Every equilateral triangle is an isosceles triangle.
  3. Woodrearrange— yours. "Every p is a q" as a picture means the p-circle sits ______ the q-circle.
  4. Earthcollapse— yours. Which two options nest the circles? Which of those two has the right one inside?
  5. WatermeaningOption C is #8's wrong answer A, drawn. The same control catches it: the picture of the converse is not the picture of the statement.
Nudge

"Every equilateral triangle is isosceles." Which circle has to be entirely inside the other — and does that tell you which is bigger?

Check my answer

B

The work, one step at a time
  1. "If p then q" is "every p is a q." Every equilateral triangle is isosceles.
  2. "Every p is a q" means the p-set sits inside the q-set. Equilateral inside Isosceles.
  3. The inner circle is the smaller set. There are fewer equilateral triangles than isosceles ones, which matches.
  4. Only B draws that. C draws the converse.
#1 · key B
  • A reads "if" as "some" — partial overlap.
  • C the converse, drawn. Same error as #8-A; if they get #8 right and #1 wrong with C, the control is understood in words and not in pictures — that gap is worth naming out loud.
  • D read "is isosceles" as "is a different kind of thing."
Mother: #8 (the converse in words). Faded: Wood and Earth were left blank on purpose; if the student cannot fill them, open The work to step 2 only.
0:42 – 0:50

What follows — and a broken solution

Third move: chain two statements. If you know p → q and you know p, you may conclude q. The test calls the result a logical conclusion. The trap is that the wrong answers are usually true. True is not the question. Follows is the question.

No scaffolding this time. Read it, answer it, then check.

SOL 2015 · #6 Reasoning, Lines, and Transformations

Given statements:
If a shape is a parallelogram, then opposite angles are congruent.
A rhombus is a parallelogram.
Which is a logical conclusion from the given statements?

Nudge

Which sentence is the p → q? Which sentence hands you a p? Then say the q with the rhombus's name in it.

Check my answer

A

The work, one step at a time
  1. Find the conditional. p: a shape is a parallelogram. q: its opposite angles are congruent.
  2. Find the fact. "A rhombus is a parallelogram" is p, said about a rhombus.
  3. Detach. p → q and p give q: a rhombus's opposite angles are congruent.
  4. Reject the rest on principle, not on truth. B and D are true; C is false; none of them was derived from the two sentences you were given.
#6 · key A
  • B true, not derived — imported a rhombus fact.
  • C false (diagonals of a rhombus are perpendicular bisectors, congruent only in a square) — a student who picks it is answering "what do I remember about rhombuses," not reading the givens.
  • D true, not derived.
Control: "true" is not "follows." This is the most important control in the whole first category. Mother: §3 (naming p and q).

A solution that goes wrong

Here is how a student answered question 6. Read the three steps. One of them is where it breaks. Click that step.

  1. 1A rhombus is a parallelogram. That is given.
  2. 2Parallelograms have opposite sides that are congruent.
  3. 3So the opposite sides of a rhombus are congruent — answer B.
The break is step 2Step 2 is a true sentence that was not one of the givens. Step 3 then reasons correctly from a premise nobody supplied. If the student clicks step 3, ask: "is step 3 wrong given step 2?" — it is not, and that is the lesson: valid reasoning from an imported premise still fails the question. Booth et al.: having the student explain the wrong step does more than showing the right one.
0:50 – 0:57

Build the argument

Last one is the test's drag-and-drop, and it is the one place today where the symbols do work for you instead of the other way round. The pattern is one you have not seen yet this hour: p → q, and then not q. What may you conclude?

SOL 2015 · #3 Reasoning, Lines, and Transformations · drag-and-drop

Let p represent Brent works this summer.
Let q represent Brent takes a vacation.
Symbolically represent the following argument.
If Brent works this summer, then he will not take a vacation.
Brent takes a vacation.
Therefore, Brent does not work this summer.

Nudge

Line 1: which of p, q is being denied by the word not? Line 2 is a plain fact — is it p or q, and is it denied? Line 3 begins with therefore.

Check my answer

p → ~q  ·  q  ·  ∴ ~p

The work, one step at a time
  1. Line 1. "If Brent works, then he will not take a vacation." Hypothesis p, conclusion not-q: p → ~q.
  2. Line 2. "Brent takes a vacation." That is q, affirmed: q.
  3. Line 3. "Therefore, Brent does not work." ∴ ~p.
  4. Why it is valid. Line 2 denies line 1's conclusion. Denying the conclusion lets you deny the hypothesis — that is the contrapositive from §3 doing its job.
#3 · key p → ~q · q · ∴ ~p Watch for: line 1 as p → q (missed the not); line 2 as ~q (read "takes a vacation" as a denial because line 1 was negative); line 3 as ∴ p (lost the not in "does not work"). The argument form is modus tollens; you need not say so, but the student should hear that this is the contrapositive in use. Mother: §3, the contrapositive check.
0:57 – 1:00

Close, and the seed

Water — what today meant. Four questions, and every one of them was the same three-word rule: the parts matter. Name the parts; swap them only when you mean to; chain them only from what you were given. That rule is not a logic-unit trick. It is how every theorem for the rest of the month will be written, and how every proof question will try to catch you.

Next time. Here is a theorem in the same shape: If two parallel lines are cut by a transversal, then alternate interior angles are congruent. Say its converse out loud before you leave. Next session that converse is the whole hour — it is what questions 5, 11, 12 and 15 are quietly testing.

SayTell me the converse of the parallel-lines statement. — Good. Is it true? We find out on Monday.
End of hourPress Summary in the strip to copy today's record — placement answers, every choice made on the four items, marks satisfied, and your notes — and put it on the bus or in a tablet. The record is the product.
Today's record